Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,11)(8,9)(10,12)(21,22,26,25)(23,29,30,27)(24,28,32,33)(31,35,36,34);(1,2,4,3)(5,6,8,9)(7,10,12,11)(14,18)(19,20)(23,30)(24,32)(27,29)(28,33);(1,3,4,2)(6,9)(10,11)(13,14)(15,19)(16,20)(17,18)(21,23,26,30)(22,27,25,29)(24,31,32,36)(28,34,33,35);(5,6)(7,10)(8,9)(11,12)(13,15,17,16)(14,19,18,20)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36);(5,7)(6,10)(8,12)(9,11)(13,16,17,15)(14,20,18,19)(21,25,26,22)(23,29,30,27)(24,33,32,28)(31,35,36,34)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=gi,d^2=gh,e^2=fh,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 460903);